624 ON THE PROBLEM OF RANDOM VIBEATIONS, AND [441
We may now seek the form approximated to when n is very great. Setting for brevity I = 1 in (59), we have
fsinx\n ( or , , . , -, , . ' }
----- =n\--a+ hos*+ kea?+ ...[,-
V x I . \ 6 j
og
\ X I
and so that
.(66)
r dr TT
The expression for the principal term is a known definite integral, and we obtain for it •
~7 == jp~ 6 , ........................\\) ( )
which may be regarded as the approximate value when n is very large. To restore I, we have merely to write r/l for r throughout.
In pursuing the approximation we have to consider the relative order of the various terms. Taking no? as standard, so that x" is regarded as of the
order Ifn, noc8 is of order n~3 and is omitted......But nzxs is of order n~2 and is
retained. The terms written down in (66) thus suffice for an approximation to the order ?r~2 inclusive. ...
The evaluation of the auxiliary, terms in (66) can be effected by differentiating the principal term with respect to n. Each such differentiatioji brings in — #2/6 as a factor, and thus four operations suffice for the inclusion of the term containing a;8.1 • • We get
dPn==^\/Q2^rN+nh Q*dW_nk 63*W + in%a 64^T| (68) dr \/TT . Is [_ cfoi" dns dn* \'
where
iv —n e ...............................(69)
Finally
_ __ _ _
dr ~ >*~' ^^ \ nl*
sir8
i
J ' ' ' '^ '
Here dPn/dr . dr is the chance that the resultant of a large number n of flights shall lie between r and r + dr. In Pearson's notation,
The maximum value of the principal term (67) occurs when r/l= £ j o x &